Differential Forms and Hodge Structures on Singular Varieties

TL;DR

This paper explores differential forms on singular algebraic varieties, relating them to Hodge structures and introducing quasi-rational singularities characterized by specific cohomological properties.

Contribution

It introduces the concept of quasi-rational singularities and connects various notions of differential forms with Hodge filtration in singular varieties.

Findings

Quasi-rational singularities are normal and relate to Du Bois and rational singularities.

An isolated singularity is rational iff it is quasi-rational, Du Bois, and certain Hodge numbers vanish.

The study links differential forms, singularities, and Hodge theory in algebraic geometry.

Abstract

We compare a couple of notions of differential form on singular complex algebraic varieties, and relate them to the outermost associated graded spaces of the Hodge filtration of ordinary and intersection cohomology. In particular, we introduce and study singularities, that we call quasi-rational, which are normal and such that for all p, the zeroth cohomology sheaf of the complex of Du Bois p-forms is isomorphic to the direct image of p-forms from a desingularization. We show that an isolated singularity is rational if and only if it is quasi-rational, Du Bois, and certain Hodge numbers of the local mixed Hodge structures vanish.